 This is a video about using the binomial probability tables in order to answer questions about the binomial distribution. In your formula book, you'll see a series of tables that look something like this. The purpose of the video is to show you how to use them. First of all, why do we need probability tables when there's a formula for calculating binomial probabilities? The reason is that some questions are very fiddly to answer using just the formula. For example, supposing that we've got a random variable which has the binomial distribution with 12 trials and 0.35 as the probability of success. And suppose we want to know the probability that x is less than or equal to 3. That's the probability of getting 3 or fewer successes. The only way of calculating this using the formula is to add together the probabilities for 0, 1, 2 and 3 successes. The formula can tell us each of those probabilities separately. But we need to work out 12 choose 0, 12 choose 1, 12 choose 2 and 12 choose 3. 12 choose 0 will be the leftmost number in a row of Pascal's triangle and all those numbers are 1, so 12 choose 0 is 1. 12 choose 1 will be the next number along and that's bound to be 12. To work out 12 choose 2, we need to do 12 factorial over 2 factorial times 10 factorial. But remember, there'll be a lot of cancelling out in that sum, so what we really need to do is 12 times 11 over 2 times 1, which is 66. 12 choose 3 will be 12 factorial over 3 factorial times 9 factorial and after the cancelling, that will be 12 times 11 times 10 over 3 times 2 times 1, which is 220. Now we can calculate the probability of no successes. We can tell that it's 1 times 1 times 0.65 to the power of 12, which is 0.00568801. We can also calculate the probability of 1 success and 2 successes and 3 successes. Finally, having calculated all 4 of these probabilities, we can add them together to find the probability that x is less than or equal to 3. It turns out to be 0.3467. Now I hope that that example was enough to convince you that using the binomial probability formula can be a real pain. The point of the binomial probability tables is that they enable us to answer questions like this one much, much faster. So how do you use the binomial probability tables? Let's see how we would use the tables to produce the answer we obtained some moments ago. First of all, we need to find the right page of the tables. We're looking for the one where it says n equals 12 because we're dealing with the binomial distribution with 12 trials. Having found that table, we need to find the column headed by 0.35 because that's the probability of success in this example. Now the question was asking us for the probability that x is less than or equal to 3. So we find the row that begins with the number 3 and follow it along until we find the probability in the right column. And you'll see there the number 0.3467. Well, this number is telling us exactly what we're looking for. It's the probability that x is less than or equal to 3. So there's our answer straight away. This shows how much simpler it can be to answer questions using the probability tables than using the formula. Unfortunately, it's not always quite so simple. So we're going to look at some more examples to see how we can use the table in other cases. First of all, suppose that x has the binomial distribution with 15 trials and 0.3 is the probability of success. What's the probability that x is greater than or equal to 5? This question is different because it's asking us about the probability of x being greater than or equal to something. Remember, the probability tables are telling us about the chance of getting less than or equal to something. So let's think, if x is greater than or equal to 5, it's 5 or more. And this is the opposite of being 4 or less. So we can work out the probability that x is greater than or equal to 5 by doing one take away the probability that x is less than or equal to 4. We can find this in the probability tables. This time we need to look for where n is equal to 15 and p is equal to 0.3. We find the row that begins with a 4 and follow it along until we hit the right column. This tells us that the probability that x is less than or equal to 4 is 0.5155. So returning to our question, we need to do the sum 1 take away 0.5155 and that gives us the answer 0.4845. Obviously that answer is accurate only to 4 decimal places. Here's another example. This time we've got a binomial distribution, again with 15 trials, but the probability of success is 0.45. We're going to work out the probability that x is greater than or equal to 6 and less than 10. Again, we need to start by thinking carefully about what numbers x could actually be in this situation. If it's greater than or equal to 6 and less than 10, then it could be 6, 7, 8 or 9. In other words, it's all the numbers from 0 up to 9 except for 01234 or 5. So the probability that x is greater than or equal to 6 but less than 10 is the probability that x is less than or equal to 9 take away the probability that it's less than or equal to 5. Okay, we can work out these two probabilities using the table. We're still looking at when n is equal to 15, but this time we're looking at the column where p is 0.45. First of all, the probability of x being less than or equal to 9, we find by following along the 9 row, and that probability is 0.9231. And secondly, the probability that x is less than or equal to 5, we find by following along the 5 row, and that's 0.2608. So the sum we need to do is 0.9231 take away 0.2608, which is 0.6623. Here's a third example. This time we want to know the probability that x is equal to something. Suppose that x has the binomial distribution with 15 trials and 0.25 is the probability of success. What's the probability that x is equal to 7? Well, we can use the same method as before. We want to know the probability that x is 7, but that's the same as the probability that x is less than or equal to 7. Take away the probability that it's less than or equal to 6. So we use the same table as before where n is equal to 15. We find the column headed by 0.25. We follow along the 7 row to get 0.9827 and the 6 row to get 0.9434. And then we do the subtraction 0.9827 take away 0.9434 to get the answer 0.0393. The next example is particularly important because it deals with a really awkward situation. Suppose that x has the binomial distribution with 15 trials and 0.8 as the probability of success. We're going to work out the chance that x is less than or equal to 10. The fact that the probability of success is 0.8 here is a real problem because you've probably noticed that in the tables p only goes up to 0.5. So what we need to do in this situation is to think about a new random variable, y, which is equal to the number of failures when there are x successes. It's equal to 15 take away the number of successes, which gives us the number of failures. The reason why this is useful is because the number of failures has the binomial distribution with 15 trials and 0.2 as the probability. Obviously, if the probability of success is 0.8, then the probability of failure is 0.2. And we can find out probabilities for y, the number of failures using the table. Now think about it, the probability that x is less than or equal to 10, the chance of getting 10 or fewer successes is the same as the chance that y is greater than or equal to 5, the chance of 5 or more failures. 10 successes is 5 failures and fewer than 10 successes is more than 5 failures. Okay, and the probability that y is greater than or equal to 5 is the same as 1 take away the probability that y is less than or equal to 4. And this is a probability that we can look up directly in the tables. We check for where n is equal to 15, we look for the column headed by 0.2, and we follow along from 4 to find the probability 0.8358. So the sum that we need to do is 1 take away 0.8358 and that's 0.1642. As before, this answer is only accurate to four decimal places. Okay, here's one last example which puts together everything we've done so far. Suppose that x has the binomial distribution with 25 trials and 0.6 as the probability of success. We're going to work out the probability that x is greater than or equal to 11 and less than 16. Well, this is a situation where the probability of success is greater than a half. So we're going to have to think about the number of failures instead. So let's set y equal to 25 minus x so that y is the number of failures. Then y will have the binomial distribution with 25 trials and 0.4 as the probability. The probability that x is greater than or equal to 11 and less than 16 is the probability that y is less than or equal to 14 and greater than 9. 11 successes is 14 failures and 16 successes is 9 failures. We would normally write that probability as the chance that y is greater than 9 and less than or equal to 14. That's going to be one probability take away another one and we better think carefully about what those probabilities are. If y is greater than 9 and less than or equal to 14, then it's 10, 11, 12, 13 or 14. That's all the numbers that are 14 or less. Take away the numbers that are 9 or fewer. So the probability that y is greater than 9 and less than or equal to 14 is the chance that y is less than or equal to 14. Take away the chance that it's less than or equal to 9. Okay, this time we look for the table where n is 25. We look for where p is 0.4. We follow along from 14 to get the probability 0.9656. From 9 to get the probability 0.4246 and then we can finish off our calculation. We need to do 0.9656. Take away 0.4246 and that gives us the answer 0.5410. Okay, there are a few things you need to bear in mind when you use the binomial probability tables. The first thing is that they give you the cumulative probability, the chance that x is less than or equal to something. Secondly, they only have information for some values of n and p. So if you have awkward numbers like 13 trials or 0.32 as the probability of success, then you'll have to use the method that I showed you at the beginning of the video. Finally, if the probability of success is greater than 0.5, then you need to think about the number of failures instead of the number of successes. That's the end of this video about using the binomial probability tables in order to answer questions about the binomial distribution.